For a symmetric, pseudo commutative, strong 2-monad T on a 2-category with finite products, the free-algebra assignment A to TA extends from a 2-functor to a Cat-enriched multifunctor into T-alg, the multicategory of algebras and pseudo maps. That multifunctor does not strictly preserve the symmetric-group action on multilinear maps, but it preserves it up to coherent invertible 2-cells built from the pseudocommutativity modification, making it pseudo symmetric in Yau's sense. Learning this means seeing how a coherence theorem is assembled: the 2-cells omega_sigma indexed by permutations, their compatibility via the Yang-Baxter and equivariance axioms, and why this settles the coherence conjectures of Hyland and Power for both symmetric and non-symmetric pseudo commutative 2-monads.
This 2025 arXiv paper by Diego Manco directly addresses coherence in 2-dimensional monad theory, specifically for pseudo commutative 2-monads. The paper proves that the free algebra functor for a sym…